Advances in Pure Mathematics, 2012, 2, 15 http://dx.doi.org/10.4236/apm.2012.21001 Published Online January 2012 (http://www.SciRP.org/journal/apm) Real Hypersurfaces in 2 and 2 Equipped with Structure Jacobi Operator Satisfying =llL n Konstantina Panagiotidou, Philippos J. Xenos Mathematics Division, School of Technology, Aristotle University of Thessaloniki, Thessaloniki, Greece Email: {kapanagi, fxenos}@gen.auth.gr Received June 27, 2011; revised August 4, 2011; accepted August 20, 2011 ABSTRACT Recently in [1], Perez and Santos classified real hypersurfaces in complex projective space 3n for , whose Lie derivative of structure Jacobi operator in the direction of the structure vector field coincides with the covariant deriva tive of it in the same direction. The present paper completes the investigation of this problem studying the case n = 2 in both complex projective and hyperbolic spaces. Keywords: Real Hypersurfaces; Complex Projective Space; Complex Hyperbolic Space; Lie Derivative; Structure Jacobi Operator 1. Introduction A complex ndimensional Kaehler manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by n c n . A complete and simply connected complex space form is complex analytically isometric to a complex projective space a complex Euclidean space n or a complex hy perbolic space P n if , or c respec tively. >0c=0 <0 n c The study of real hypersurfaces was initiated by Ta kagi (see [2]), who classified homogeneous real hyper surfaces in n and showed that they could be divided into six types, which are said to be of type A1, A2, B, C, D and E. Berndt (see [3]) classified homogeneous real hy persurfaces in n P n with constant principal curvatures. Okumura (see [4]) in and Montiel and Romero (see [5]) in gave the classification of real hyper surfaces satisfying relation =0AA . Ki and Liu (see [6]) have given the above classifica tion as follows: Theorem A Let M be a real hypersurface of n c 0cA , , (n ≥ 2). If it satisfies=0A n , then M is lo cally congruent to one of the following hypersurfaces: In case 1 a geodesic hypersphere of radius r, where π 0<< 2 r , 2 a tube of radius r over a totally geodesic k , 2kn1 , where π 0< <. 2 r n In case n 0 a horosphere in , i.e. a Montiel tube, 1 a geodesic hypersphere or a tube over a hyper plane 1n , k 2 a tube over a totally geodesic 12kn n . c0c , (Let M be a real hypersurface in ). Then an almost contact metric structure (,,, ) can be defined on M. The structure vector field is called principal if =A holds on M, where A is the shape operator of M in n c and is a smooth function. A real hypersurface is said to be a Hopf hypersurface if is principal. The Jacobi operator field with respect to X on M is de fined by ,RXX =X , where R is the Riemmanian curva ture of M. For the Jacobi operator is called structure Jacobi operator and is denoted by =,lR . It has a fundamental role in almost contact manifolds. Many differential geometers have studied real hypersur faces in terms of the structure Jacobi operator. The Lie derivative of the structure Jacobi operator with respect to was investigated by Perez, Santos, Suh (see [7]). More precisely, they classified real hyper surfaces in n 3 =0l 2 (n), whose structure Jacobi op erator satisfies the condition: L. Ivey and Ryan (see [8]) classified real hypersurfaces satisfying the same condition in 2 and . The study of real hypersurfaces whose structure Jacobi operator is parallel is a problem of great importance. In [9] the nonexistence of real hypersurfaces in nonflat C opyright © 2012 SciRes. APM
K. PANAGIOTIDOU ET AL. 2 ,= ,, ,= , complex space form with parallel structure Jacobi opera tor () was proved. In [10] a weaker condition (parallelness), X for any vector field X or thogonal to XY gXYXY gX YgXY =, =, X X AX YYAXgAXY =0l =0l , was studied and it was proved the non existence of such hypersurfaces in case of n 3 n (n). The parallelness of structure Jacobi operator in combina tion with other conditions was another problem that was studied by many others such as Ki, Kim, Perez, Santos, Suh (see [11,12]). Recently PerezSantos (see [1]) studied real hypersur faces in for , whose structure Jacobi opera tor satisfies the relation: 3n =.ll L P H P H (1) In the present paper we go on studying the same prob lem for 2 and 2. We prove the following theo rem: Main Theorem Let M be a real hypersurface in 2 or 2, whose structure Jacobi operator satisfies rela tion (1). Then M is locally congruent to: a geodesic sphere of radius r, where π <2 r0< with π 4 r, or to a tube of radius π =4 r over a holomorphic curve in 2 and to a horosphere, a geodesic sphere or a tube over P 1 A in 2i or to a Hopf hypersurface in 2 with HH =0 . 2. Preliminaries Let M be a connected real hypersurface immersed in a nonflat complex space form , n cG 0c = , (), with almost complex structure J of constant holomorphic sec tional curvature c. Let N be a unit normal vector field on M and N . For a vector field X tangent to M we can write = XX XN , where φX and N are the tangential and the normal component of JX re spectively. The Riemannian connection in n c and in M are related for any vector fields X, Y on M: =, YY XgAYXN = XNAX where g is the Riemannian metric on M induced from G of n c and A is the shape operator of M in n c (,,) . M has an almost contact metric structure in duced from J on n c where is a (1,1) tensor field and is a 1form on M such that ,= ,, XY G JXY = , =, gX GJXN =0,X (see [13]). Then we have 2=, =0, =1 XX X (2) (3) (4) Since the ambient space is of constant holomorphic sectional curvature c, the equations of Gauss and Codazzi for any vector fields X, Y, Z on M are respectively given by ,= ,,, 4 ,2, ,, c RXYZgYZXgXZY gYZX gXZYgXYZ gAYZAXgAXZAY (5) =4 2, XY c YAX XYYX gXY PM (6) where R denotes the Riemannian curvature tensor on M. For every point , the tangent space can be decomposed as following: P TM = P T Mspanker where =:=0kerXTMX P. Due to the above decomposition, the vector field is decom posed as follows: U= (7) where = and 1 =U ker 0 , pro vided that . All manifolds are assumed connected and all mani folds, vector fields etc are assumed smooth (C ). 3. Auxiliary Relations 2 2 or Suppose now that the ambient space is , (i.e. 2c0c, ), then we consider V be the open subset of points P M, such that there exists a neighbor hood of every P, where =0 and the open subset of points Q of M such that there exists a neighborhood of every Q, where 0a . Since, is a smooth function on M, then is an open and dense subset of M. V Proposition 3.1 Let M be a real hypersurface in 2 c, equipped with structure Jacobi operator satis fying (1). Then, is principal on V. A Proof: The relation (7) on V, takes the form U UYZ . From (5) for , we obtain: . 4 c lU U (8) Copyright © 2012 SciRes. APM
K. PANAGIOTIDOU ET AL. 3 Due to the definition of Lie derivative, the relation (1) for X yields: l 0 0 . The latter, because of the first relation of (4) and (8) implies A , hence 0 . Therefore, is principal on V. □ On if =0 , then is principal. In what fol lows we work on W (W), which is the open subset of points such that 0 Q . Lemma 3.2 Let M be a real hypersurface in 2 c. Then the following relations hold on W: 2 , 44 cc UU AUU (9) 2 , 44 UU cc UU , U (10) 12 ,, 4 UU c UUU U 3 UU (11) 2 23 , UU 14 , U U c UU UU 3 ,, ,,UU (12) where are smooth functions on M. 12 Proof: If =,=, is an orthonormal basis, then because of (7) we have: UU UAUUU (13) where , and are smooth functions on M. The first relation of (4), because of (13) yields: =, =. U U = , U UU UU (14) The relation (5), using (13) can be written: 2 4 c lU lU U , . 4 U U cU . lX X l (15) By the definition of Lie derivative, the relation (1) takes the form: The latter for X ,U 0lU implies lU = 0, and then from (15) we obtain: 2 44 cc 0, ,. , XX (16) The relations (13) and (14), because of (16) imply (9) and (10) respectively. From the well known relation: ,, gYZgYZ ,, gY Z for YZ ,,UU , using (16) we obtain (11) and (12), where 1 , 2 and 3 are smooth functions on M. □ ,,UU The Codazzi equation for X, Y , because of Lemma 3.2 yields: 2 2 41,Uc (17) 22 3 1, 44 cc (18) 2 2 4,Uc (19) 2 2 4, c (20) 3 3, 4 c U (21) 2 2 1, 24 cc U (22) 22 13. 44 cc U (23) The Riemannian curvature on M satisfies (5) and on the other hand is given by the relation ,RXYZZZ Z ,,UU , XY YXXY . From these two relations and because of (16) for X, Y we obtain: 2 3 21 2 22 3 12 242 c cc UU c (24) 2 312 3 4 c U (25) 3213 3 42 cc U 34 (26) Relation (23), because of (18), (21) and (22), yields: (27) and so relation (18) becomes: 2 2 14. 44 cc (28) Differentiating the relations (27) and (28) with respect to U and respectively and substituting in (25) and due to (19), (20) and (27) we obtain: 22 224 0c . (29) Copyright © 2012 SciRes. APM
K. PANAGIOTIDOU ET AL. 4 Owing to (29), we consider , (W) the open subset of points , where in a neighbor hood of every Q. 1 W1W 20 QW Lemma 3.3 Let M be a real hypersurface in 2 c 1 W 2 c , equipped with Jacobi operator satisfying (1). Then is empty. Proof: Due to (29) we obtain: on 1 W. Differentiation of the last relation along 2 24 and taking into account (19), (20) and yields: , which is a contradiction. Therefore, is empty. □ 2 24 2 c =0c 2=0 0. 1 On W, because of Lemma 3.3, we have , hence the relations (17), (19) and (20) become: W UU Using the last relations and Lemma 3.2 we obtain: ,0,U UU .cU 22 , 1416 4 U UU 0cU WWQW Combining the last two relations we have: 22 416 . (30) Let 2, (2) be the set of points W , for which there exists a neighborhood of every Q such that . So from (30) we have: 16 . Differentiating the last relation with respect to U 0 22 4c U and taking into account (21), (22), (27) and (28), we have: , which is impossible. So 2 is empty. Hence, on W we have . Then, relations (21) and (28), because of (27) imply respectively: 20 W U 0 2 4c 2 and 2 5 1 . Relation (26), because of 0U , 2 4c and (27) yields: 1 2 . Substitution of 1 in 1 2 52 yields: 322 . Differentiation of the last one along U and taking into account (22) leads to: 0 , which is a contradiction. So we obtain the following proposition: Proposition 3.4 Let M be a real hypersurface in 2 c, equipped with Jacobi operator satisfying (1). Then M is a Hopf hypersurface. 4. Proof of Main Theorem Since M is a Hopf hypersurface, we can write ZZ and ZZ with ,, Z , being a local or thonormal basis and the following relation holds: 24 c (Corollary 2.3 [14]). Furthermore, due to Theorem 2.1 [14], we have that is constant. From (5), due to ZZ and ZZ we get: ,. 44 cc lZZl ZZ (31) The relation (1), because of (31) implies: 0 , for Z and , for 0 Z . Combining the last two relations leads to: 20 2 , M is locally 0 . If , in case of congruent to a tube of radius π 4 2 over a holomorphic curve in 0A due to [15] and to a Hopf hypersurface with 2 . If in 0 , the last relation im plies: and we obtain: 0, , AX XTM which because of Theorem A completes the proof of Main Theorem. 5. Acknowledgements The first author is granted by Foundation Alexandros S. Onasis. Grant Nr: G ZF 044/20092010. The authors would like to thank Professors Juan de Dios Perez and Patrick J. Ryan. Their answers improved the proof of main theorem. Finally, the authors express their grati tude to the referee who has contributed to improve the paper. REFERENCES [1] J. D. Perez and F. G. Santos, “Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Op erator Satisfies R ,” Rocky Mountain Journal of Mathematics, Vol. 39, No. 4, 2009, pp. 12931301. L doi:10.1216/RMJ20093941293 [2] R. Takagi, “On Homogeneous Real Hypersurfaces in a Complex Projective Space,” Osaka Journal of Mathe matics, Vol. 10, 1973, pp. 495506. [3] J. Berndt, “Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Space,” Journal fur die Reine und Angewandte Mathematik, Vol. 395, 1989, pp. 132141. doi:10.1515/crll.1989.395.132 [4] M. Okumura, “On Some Real Hypersurfaces of a Com plex Projective Space,” Transactions of the American Mathematical Society, Vol. 212, 1975, pp. 355364. doi:10.1090/S0002994719750377787X [5] S. Montiel and A. Romero, “On Some Real Hypersur faces of a Complex Hyperbolic Space,” Geometriae Dedi cata, Vol. 20, No. 2, 1986, pp. 245261. doi:10.1007/BF00164402 [6] U.H. Ki and H. Liu, “Some Characterizations of Real Hypersurfaces of Type (A) in Nonflat Complex Space Form,” KMSBulletin of the Korean Mathematical Society, Vol. 44, No. 1, 2007, pp. 157172. doi:10.4134/BKMS.2007.44.1.157 [7] J. D. Perez, F. G. Santos and Y. J. Suh, “Real Hypersur Copyright © 2012 SciRes. APM
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